I am not sure if Cramer's rule is used for computation purposes. Your help would mean a lot. Thanks!
Cramer is highly inefficient, of time complexity $O(n! \times n)$ with a naive determinant-finding algorithm, and $O(n^4)$ with e.g. LU decomposition. Gaussian elimination has cubic complexity.
Reducing the matrix to triangular form and multiplying the elements on the diagonal is usually quicker. I am pretty sure that is the algorithm most computer algebra systems use, unless it is known in advance that the matrix has some special properties. Sometimes Laplace expansion can be quicker if the matrix has many zeros along some rows/columns.